1、2,第五章 軸向拉伸和壓縮,CHAPTER 5 AXIAL TENSION AND COMPRESSION,1,3,51 軸向拉壓的概念及實例 52 內力、截面法、軸力及軸力圖 53 截面上的應力及強度條件,第五章 軸向拉伸和壓縮,5-4 拉壓桿的變形 彈性定律,5-5 拉壓桿的彈性應變能,5-6 拉壓超靜定問題及其處理方法,5-7 材料在拉伸和壓縮時的力學性能,CHAPTER 5 AXIAL TENSION AND COMPRESSION,51 Concepts and practical examples of axial tension and compression 52 Intern

2、al force、method of section、axial force and its diagram 53 Stresses on the section and strength conditions,5-4 Deformation of the rod in axial tension and compression law of elasticity,5-5 Elastic strain energy of the rod in axial tension and compression,5-6 Statically indeterminate problems and thei

3、r treatment methods of axial tension and compression,5-7 Mechanical properties of materials in axial tension and compression,3,5,拉壓,51 軸向拉壓的概念及實例,軸向拉壓的外力特點：外力的合力作用線與桿的軸線重合。,一、概念,軸向拉壓的變形特點：桿的變形主要是軸向伸縮，伴隨橫向 縮擴。,軸向拉伸：桿的變形是軸向伸長，橫向縮短。,軸向壓縮：桿的變形是軸向縮短，橫向變粗。,51 CONCEPTS AND PRACTICAL EXAMPLES OF AXIAL TENSI

4、ON AND COMPRESSION,Characteristic of the external force：The acting line of the resultant of external forces is coincided with the axis of the rod.,1、Concepts,Characteristic of the deformation：Deformation of the rod is mainly elongation or contraction along the axis of the rod and companied with late

5、ral reduction or enlargement.,Axial tension ：Deformation of the rod is axial elongation and lateral shortening.,Axial compression：Deformation of the rod is axial shortening and lateral enlargement.,AXIAL TENSION AND COMPRESSION,5,7,拉壓,軸向壓縮，對應的力稱為壓力。,軸向拉伸，對應的力稱為拉力。,力學模型如圖,In axial compression，the cor

6、responding force is called compressive force.,In axial tension，the corresponding force is called tensile force.,Mechanical models are shown in the figures,AXIAL TENSION AND COMPRESSION,7,9,拉壓,Practical examples in engineering,2、,AXIAL TENSION AND COMPRESSION,9,11,拉壓,AXIAL TENSION AND COMPRESSION,11,

7、13,討論：下列桿件哪些是軸向受拉桿？,14,Discussion: which are axial rods of tension ?,15,拉壓,一、內力 指由外力作用所引起的、物體內相鄰部分之間分布內力系的合成（附加內力）。,52 內力 截面法 軸力及軸力圖,1、Internal force Internal force is the resultant of internal forces, which is acting mutually between two neighbour parts inside the body,caused by the external forces

8、.,52 INTERNAL FORCE、METHOD OF SECTION、AXIAL FORCE AND ITS DIAGRAM,AXIAL TENSION AND COMPRESSION,13,17,拉壓,二、截面法 軸力,內力的計算是分析構件強度、剛度、穩(wěn)定性等問題的基礎。求內力的一般方法是截面法。,1. 截面法的基本步驟： 截開：在所求內力的截面處，假想地用截面將桿件一分為二。 代替：任取一部分，其棄去部分對留下部分的作用，用作用 在截開面上相應的內力（力或力偶）代替。 平衡：對留下的部分建立平衡方程，根據(jù)其上的已知外力來 計算桿在截開面上的未知內力（此時截開面上的內力 對所留部分而言

9、是外力）。,2、Method of section axial force,Calculation of the internal forces is the foundation to analyze the problems of strength、rigidity、stability etc. The general method to determine internal forces is the method of section.,1). Basic steps of the method of section： Cut off：Assume to separate the ro

10、d into two distinct parts in the section in which the internal forces are to be determined. Substitute：Take arbitrary part and substitute the action of another part on it by the corresponding internal force in the cut-off section. Equilibrium：Set up equilibrium equations for the remained part and de

11、termine the unknown internal forces according to the external forces acted on it.（Here the internal forces in the cut-off section are the external forces for the remained part）,AXIAL TENSION AND COMPRESSION,15,19,拉壓,2. 軸力軸向拉壓桿的內力，用N 表示。,例如： 截面法求N。,截開：,代替：,平衡：,2. Axial forceinternal force of the rod

12、in axial tension or compression，designated by N .,Such as： Determine N by the method of section.,Cut off:：,Substitute：,Equilibrium：,AXIAL TENSION AND COMPRESSION,17,21,反映出軸力與橫截面位置變化關系，較直觀； 確定出最大軸力的數(shù)值 及其所在橫截面的位置， 即確定危險截面位置，為 強度計算提供依據(jù)。,拉壓,三、 軸力圖 N (x) 的圖象表示。,3. 軸力的正負規(guī)定:,N 與外法線同向,為正軸力(拉力),N與外法線反向,為負軸力(

13、壓力),N,x,P,意義,Reflect the variety relation between the corresponding axial force and the position of the section. Find out value of the maximum axial force and the position of the section in which the maximum axial force act. That is to determine the position of the critical section and supply the in

14、formation for the calculation of strength.,3、 Diagram of the axial force sketch expression of N (x),3). Sign conventions for the axial force:,axial force N (tensile force)is positive when its direction point to the outward direction of the normal line of the section, (compressive force)negative inwa

15、rd,x,meaning,AXIAL TENSION AND COMPRESSION,19,23,拉壓,例1 圖示桿的A、B、C、D點分別作用著大小為5P、8P、4P、 P 的力，方向如圖，試畫出桿的軸力圖。,解： 求OA段內力N1：設置截面如圖,Example 1 The forces with magnitudes 5P、8P、4P and P act respectively at points A、B、C、D of the rod. Their directions are shown in the figure. Try to plot the diagram of the axia

16、l force of the rod.,Solution： Determine the internal force N1 in segment OA. Take the free body as shown in the figure.,AXIAL TENSION AND COMPRESSION,21,25,拉壓,同理，求得AB、BC、CD段內力分別為：,N2= 3PN3= 5P N4= P,軸力圖如右圖,D,PD,N,x,2P,3P,5P,P,Similarly，we get the internal forces in segment AB、BC、CD . They are respec

17、tively：,N2= 3PN3= 5P N4= P,The diagram of the axial force is shown in the right figure .,D,PD,N,x,2P,3P,5P,P,AXIAL TENSION AND COMPRESSION,23,27,拉壓,軸力(圖)的簡便求法： 自左向右:,軸力圖的特點：突變值 = 集中載荷,遇到向左的P， 軸力N 增量為正； 遇到向右的P ， 軸力N 增量為負。,3kN,5kN,8kN,Simple method to plot the diagram of axial force： From the left to

18、the right:,Characteristic of the diagram of the axial force：Value of sudden change = concentrated load,If meeting the force P to the left ，the increase of the axial force N is positive； If meeting the force to the right ，the increase of the axial force N is negative.,3kN,5kN,8kN,AXIAL TENSION AND CO

19、MPRESSION,25,29,拉壓,解：x 坐標向右為正，坐標原點在 自由端。 取左側x 段為對象，內力N(x)為：,q,q L,x,O,例2 圖示桿長為L，受分布力 q = kx 作用，方向如圖，試畫出 桿的軸力圖。,L,q(x),N,x,O,Solution：The free end of the rod is the origin of the coordinate and coordinate x to the right is positive. Take the segment of length x on the left of point x, its internal fo

20、rce is,q,K L,x,O,Example 2 Length of the rod shown in the figure is L. Distributed force q = kx is acted on it, direction of the force is shown in the figure. Try to plot the diagram of axial force of the rod.,L,q(x),N,x,O,AXIAL TENSION AND COMPRESSION,27,31,拉壓,一、應力的概念,53 截面上的應力及強度條件,問題提出：,1. 內力大小不能

21、衡量構件強度的大小。 2. 強度：內力在截面的分布集度應力； 材料承受荷載的能力。,1. 定義：由外力引起的內力集度。,1、Concept of stress,53 STRESSES ON THE SECTION AND STRENGTH CONDITIONS,Bring forward the problem：,1). The magnitude of the internal force can not scale the strength of the structure member. 2). Strength：Intensity of the distributed internal

22、 forces in the sectionstress； The load-bearing capacity of the material.,1). Definition：Intensity of the internal force due to the external forces.,AXIAL TENSION AND COMPRESSION,29,33,拉壓,工程構件，大多數(shù)情形下，內力并非均勻分布，集度的定義不僅準確而且重要，因為“破壞”或“失效”往往從內力集度最大處開始。,平均應力：,全應力（總應力）：,2. 應力的表示：,Under most cases distributi

23、on of the internal force inside engineering members is not uniform. Definition of intensity is neither accurate and important because breakage or failure often begins from the point at which intensity of the internal force is maximum.,Average stress：,Whole stress（sum stress）：,2). Expression of stres

24、s：,AXIAL TENSION AND COMPRESSION,31,35,拉壓,全應力分解為：,a.垂直于截面的應力稱為“正應力” (Normal Stress)；,b.位于截面內的應力稱為“剪應力”(Shearing Stress)。,Whole stress may be decomposed into:,a. Stress perpendicular to the section is called“normal stress”,b. Stress lying in the section is called“shearing stress”,AXIAL TENSION AND CO

25、MPRESSION,33,37,拉壓,變形前,1. 變形規(guī)律試驗及平面假設：,平面假設：原為平面的橫截面在變形后仍為平面。 縱向纖維變形相同。,受載后,二、拉（壓）桿橫截面上的應力,Before deformation,1). Experiment on the law of deformation and the hypothesis of plane section：,Hypothesis of plane section：Cross sections remain planes before and after deformations . Deformations of longitu

26、dinal fibers are the same,After loading,2、Stress in the cross section of the rod in tension or compression,AXIAL TENSION AND COMPRESSION,35,39,拉壓,均勻材料、均勻變形，內力當然均勻分布。,2. 拉伸應力：,軸力引起的正應力 ： 在橫截面上均布。,危險截面：內力最大的面，截面尺寸最小的面。 危險點：應力最大的點。,3. 危險截面及最大工作應力：,The material is homogeneous and , its deformation is un

27、iform , so the internal force is distributed uniformly。,2. Tensile stress：,Normal stress due to the axial forces ： distributes uniformly in the cross section.,Critical section：The section in which internal force is maximum and of which the dimension is smallest. Critical point：The point at which the

28、 stress is maximum.,3. Critical section and the maximum working stress ：,AXIAL TENSION AND COMPRESSION,37,41,拉壓,直桿、桿的截面無突變、截面到載荷作用點有一定 的距離。,4. 公式的應用條件：,6. 應力集中（Stress Concentration）：,在截面尺寸突變處，應力急劇變大。,5. Saint-Venant原理：,離開載荷作用處一定距離，應力分布與大小不受外載荷作用方式的影響。,Straight rod、cross section of the rod is without

29、 sudden change、there is a certain distance from the section to the point at which the load acts.,4). Application conditions of the formula：,6). Stress concentration：,Stress increases abruptly near the cross section with a sudden change in dimension,5). Saint-Venant principle：,Distribution and magnit

30、ude of the stress in the section at a certain distance from the point at which the load is acted are not affected by the acting form of external loads.,AXIAL TENSION AND COMPRESSION,39,43,拉壓,Saint-Venant原理與應力集中示意圖,(紅色實線為變形前的線，紅色虛線為紅色實線變形后的形狀。),變形示意圖：,應力分布示意圖：,Sketch of Saint-Venant principle and str

31、ess concentrations,(Red real lines denote the line before deformation and red dashed lines denote the shape after deformation.),Sketch of deformation：,Sketch of the stress distribution：,AXIAL TENSION AND COMPRESSION,41,45,拉壓,7. 強度設計準則（Strength Design）：,其中：-許用應力， max-危險點的最大工作應力。,設計截面尺寸：,依強度準則可進行三種強度計

32、算：,保證構件不發(fā)生強度破壞并有一定安全余量的條件準則。,校核強度：,許可載荷：,7). Criterion of the strength design：,where：allowable stress， max the maximum working stress at the critical point.,Design the dimension of the section：,Three kinds of calculation of strength may be done according to the criterion of strength:,That structure

33、members are ensured not to be wrecked and have certain safe degree.,Check the strength：,Determine the allowable load：,AXIAL TENSION AND COMPRESSION,43,47,拉壓,例3 已知一圓桿受拉力P =25 k N，直徑 d =14mm，許用應力 =170MPa，試校核此桿是否滿足強度要求。,解： 軸力：N = P =25kN,應力：,強度校核：,結論：此桿滿足強度要求，能夠正常工作。,Example 3 A circular rod is subject

34、ed to a tensile force P =25 kN. Its diameter is d =14mm and its allowable stress is =170MPa. Try to check the strength of the rod.,Solution： Axial force：N = P =25kN,Stress：,Check the strength：,Conclusion：The strength of the rod satisfies request. The rod can work normally.,AXIAL TENSION AND COMPRESS

35、ION,45,49,拉壓,例4 簡易起重機構如圖，AC為剛性梁，吊車與吊起重物總重為P，為使 BD桿最輕，角 應為何值？ 已知 BD 桿的許用應力為。,分析：,x,L,h,q,P,A,B,C,D,Example 4 A simple crane is shown in the figure. AC is a rigid beam ，sum weight of the hoist and heavy body that is lifted is P. What should be the angle so that the rod BD has the minimum weight？ The a

36、llowable stress of the rod is known .,Analysis：,x,L,h,q,P,A,B,C,D,AXIAL TENSION AND COMPRESSION,53,51,拉壓, BD桿橫截面面積A：,解： BD桿內力N(q )： 取AC為研究對象，如圖,YA,XA,NBD,x,L,P,A,B,C, The cross-section area A of the rod BD：,Solution： Internal force N(q ) of the rod BD： Take AC as our study object as shown in the fig

37、ure.,YA,XA,NBD,x,L,P,A,B,C,AXIAL TENSION AND COMPRESSION,55,53,拉壓,YA,XA,NBD,x,L,P,A,B,C, 求VBD 的最小值：,YA,XA,NBD,x,L,P,A,B,C, Determine the minimum value of VBD ：,AXIAL TENSION AND COMPRESSION,67,55,拉壓,三、拉(壓)桿斜截面上的應力,設有一等直桿受拉力P作用。 求：斜截面k-k上的應力。,解：采用截面法 由平衡方程：Pa=P,則：,Aa：斜截面面積；Pa：斜截面上內力。,由幾何關系：,代入上式，得：,斜

38、截面上全應力：,3、Stresses in the inclined section of the rod in tension or compression,Aa:Area of the inclined section；Pa：Internal force in the inclined section.,：,From geometric relation,Substituting it into the above formula we get,Solution: Adopt the method of section. According to the equilibrium equat

39、ion：Pa=P then,Assume a straight rod is subjected to a tensile force P. Determine the stress in the inclined section k-k .,Whole stress in the inclined section：,AXIAL TENSION AND COMPRESSION,59,57,拉壓,斜截面上全應力：,分解：,反映：通過構件上一點不同截面上應力變化情況。,當 = 90時，,當 = 0,90時，,Decomposition：,It indicates the change of str

40、esses in different sections through a point.,As =90，,As = 0,90，,Whole stress in the inclined section：,AXIAL TENSION AND COMPRESSION,61,59,2、單元體：單元體構件內的點的代表物，是包圍被研究點的 無限小的幾何體，常用的是正六面體。 單元體的性質a、平行面上，應力均布； b、平行面上，應力相等。,3、拉壓桿內一點M 的應力單元體:,1.一點的應力狀態(tài)：過一點有無數(shù)的截面，這一點的各個截面 上的應力情況，稱為這點的應力狀態(tài)。,補充：,拉壓,2、Element：El

41、ement delegate a point inside the member, infinitesimal geometric body which envelops the study point. The element in common use is just hexahedron properties of an elementa、stress is distributed uniformly in an arbitrary parallel plane；b、stresses in the parallel plane are equal.,3、stress element at

42、 a point M in the rod in tension or compression:,1. State of stress at a point：There are countless sections through a point. Sum of stresses in the different section through a point is called the state of stress at this point.,Complementary：,AXIAL TENSION AND COMPRESSION,63,61,取分離體如圖3， a 逆時針為正； t a

43、繞研究對象順時針轉為正；由分離體平衡得：,拉壓,4、拉壓桿斜截面上的應力,Take a free body as shown in the Fig.3. a is positive if it is along counterclockwise； t is positive if it makes the free body rotate clockwise. From the equilibrium of the free body we get：,4、Stress in the inclined section of the rod in tension or compression,AX

44、IAL TENSION AND COMPRESSION,65,63,例6 直徑為d =1 cm 桿受拉力P =10 kN的作用，試求最大剪應力，并求與橫截面夾角30的斜截面上的正應力和剪應力。,解：拉壓桿斜截面上的應力，直接由公式求之：,拉壓,Example 6 A rod, which the diameter d =1 cm is subjected to a tensile force P =10kN. Determine the maximum shearing stress , the normal stress and shearing stress in the inclined

45、 section of an angle 30about the cross section.,Solution：Stresses in the inclined section of the rod in tension or compression can be determined directly by the formula：,AXIAL TENSION AND COMPRESSION,67,65,例7圖示拉桿沿mn由兩部分膠合而成,受力P，設膠合面的許用拉應力為=100MPa ；許用剪應力為=50MPa ，并設桿的強度由膠合面控制,桿的橫截面積為A= 4cm，試問:為使桿承受最大拉

46、力,角值應為多大?(規(guī)定: 在060度之間)。,聯(lián)立(1)、(2)得：,拉壓,解：,Example 7 A tensile rod as shown in the figure is made from two parts glued mutually together along mn. It is subjected to the action of force P. Assume that the allowable normal stress is =100MPa and allowable shearing stress is =50MPa for the adhesive. Are

47、a of cross section of the rod is A= 4cm. If strength of the rod is controlled by the adhesive what is the angle ( : between 0 060 0) to get the largest tensile force?,Combine (1)、(2) and get：,Solution：,AXIAL TENSION AND COMPRESSION,69,67,(1)、(2)式的曲線如圖(2)，顯然，B點左 側由正應力控制桿的強度，B點右側由剪應力控制桿的強度，當a=60時，由(2)

48、式得,解(1)、(2)曲線交點處：,拉壓,討論：若,The curves of formula (1)and、(2) are shown in the Tig.(2). Obviously the strength of the rod on the left of point B is controlled by the normal stress，that on the right of point B is controlled by the shearing stress. As a=60，from formula (2) we can get,Solution: At the poi

49、nt of intersection of curves (1) and (2)：,Discussion：As,AXIAL TENSION AND COMPRESSION,71,69,1、桿的縱向總變形：,3、平均線應變：,2、線應變：單位長度的線變形。,一、拉壓桿的變形及應變,54 拉壓桿的變形 彈性定律,拉壓,1)、The whole longitudinal deformation of the rod：,3)、Average stain：,2)、Strain：linear deformation per unit length.,1、Deformation and strain of

50、the rod in tension or compression,54 DEFORMATION OF THE ROD IN AXIAL TENSION AND COMPRESSION LAW OF ELASTICITY,AXIAL TENSION AND COMPRESSION,73,71,4、x點處的縱向線應變：,6、x點處的橫向線應變：,5、桿的橫向變形：,拉壓,L1,4、Longitudinal strain at point x：,6、Lateral strain at point x：,5、Lateral deformation of the rod：,L1,AXIAL TENSI

51、ON AND COMPRESSION,75,73,二、拉壓桿的彈性定律,1、等內力拉壓桿的彈性定律,2、變內力拉壓桿的彈性定律,內力在n段中分別為常量時,“EA”稱為桿的抗拉壓剛度。,拉壓,2、Elastic law of the rod in tension or compression,1)、Case of equal internal forces,2)、Case of variable internal forces,When internal forces in n segments are constant,“EA”is called the axial rigidity of t

52、he rod in tension or compression.,AXIAL TENSION AND COMPRESSION,77,75,3、單向應力狀態(tài)下的彈性定律,4、泊松比（或橫向變形系數(shù)）,拉壓,三、是誰首先提出彈性定律 彈性定律是材料力學等固體力學一個非常重要的基礎。一般認為它是由英國科學家胡克(1635一1703)首先提出來的，所以通常叫做胡克定律。其實，在胡克之前1500年，我國早就有了關于力和變形成正比關系的記載。,3)、Elastic law in uniaxial stressed state,4)、Possion , s ratio（or coefficient of

53、the lateral deformation）,3、Who firstly proposed the Elastic Law The Elastic Law is the important foundation of solid mechanics such Mechanics of Materials .Generally it is considered to be proposed firstly by the English scientist Hook (1635-1703), So the Elastic Law is also called Hook, s Law .Actu

54、ally , there was an early record of the proportional ration between force and deformation , which is 1500 years earlier than Hook.,or,AXIAL TENSION AND COMPRESSION,79,77,東漢經(jīng)學家鄭玄(127200)對考工記弓人中“量其力，有三均”作了 這樣的注釋：“假令弓力勝三石，引之中三尺，弛其弦，以繩緩擐之，每加物一石，則張一尺?！?(圖),拉壓,東漢經(jīng)學家鄭玄(127200)對考工記弓人中“量其力，有三均”作了 這樣的注釋：“假令弓力

55、勝三石，引之中三尺，弛其弦，以繩緩擐之，每加物一石，則張一尺?！?(圖),AXIAL TENSION AND COMPRESSION,79,拉壓,83,AXIAL TENSION AND COMPRESSION,81,1、怎樣畫小變形放大圖？,變形圖嚴格畫法，圖中弧線；,求各桿的變形量Li ，如圖1；,變形圖近似畫法，圖中弧之切線。,例8 小變形放大圖與位移的求法。,拉壓,1、How to plot the enlargement sketch of the small deformation,Accurate method to plot diagram of deformation，the

56、 arc line as shown in the figure；,Determine deformation Li of each rod as shown in Fig l.,Approximate method to plot the diagram of deformation; the tangent of the arc line shown in the figure.,Example 8 Enlargement sketch of the small deformation and the method to determine displacements,AXIAL TENS

57、ION AND COMPRESSION,85,83,2、寫出圖2中B點位移與兩桿變形間的關系,拉壓,解：變形圖如圖2， B點位移至B點，由圖知：,圖 2,2、Write the relation between the displacement of point B shown in Fig.2 and deformations of two rods.,Solution：The diagram of deformation is shown in the Fig.2. Point B moves to point B，F(xiàn)rom the diagram of displacement we may know：,AXIAL TENSION AND COMPRESSION,87,85,55 拉壓桿的彈性應變能,一、彈性應變能：桿件發(fā)生彈性變形，外力功轉變?yōu)樽冃文苜A存 于桿內，這種能成為應變能（Strain Energy）用“U”表示。,二、 拉壓桿的應變能計算： 不計能量損耗時，外力功等于應變能。,內力為分 段常量時,拉壓,55 ELASTIC STRAIN ENERGY OF THE ROD IN AXIAL TENSION AND COMPRESSION,1、Elastic s